Variations on Debris Disks IV. An Improved Analytical Model for Collisional Cascades

We derive a new analytical model for the evolution of a collisional cascade in a thin annulus around a single central star. In this model, $r_{max}$ the size of the largest object declines with time (t); $r_{max} \propto t^{-\gamma}$,with$\gamma$=0.1-\mathrm{0.2.}Comparedtostandardmodelswhere$r_{max}~$ is constant in time, this evolution results in a more rapid decline of $M_d$, the total mass of solids in the annulus and $L_d$, the luminosity of small particles in the annulus: $M_d\propto t^{-(\gamma +1)}$ and $L_d \propto t^{-(\gamma/2 + 1)}~~$. We demonstrate that the analytical model provides an excellent match to a comprehensive suite of numerical coagulation simulations for annuli at 1 AU and at 25 AU. If the evolution of real debris disks follows the predictions of the analytical or numerical models, the observed luminosities for evolved stars require up to a factor of two more mass than predicted by previous analytical models.